#maths #vectors 

# surfaces

$\vec{r}(s, t) = \left(x(s, t), y(s, t), z(s, t) \right)$
when using the tuple $(s, t)$ to define a surface, the surface is given by functions in $x, y, z$ of that tuple

if the surface is a rectangle, this means that $a\leq s\leq b$ and $c\leq t \leq d$
the lines $s=\text{const}$ and $t = \text{const}$ are coordinate lines on the surface

if the surface is a cylinder of radius $h$, 
$\vec{r}(s, t) = a\cos(s)\vec{i} + a\sin(s)\vec{j} + t\vec{k}$, $-\pi \leq s < \pi$, $0\leq t \leq h$
the lines $t=\text{const}$ are circles and $s=\text{const}$ are vertical lines on the surface of the cylinder

if the surface is a sphere of radius a with a circle at the origin
$\vec{r}(s, t) = a\cos s\sin t \vec{i} + a\sin s\sin t \vec{j} + a \cos t \vec{k}$, $-\pi \leq s< \pi$, $0\leq t \leq \pi$
$s=\text{const}$ are lines of longitude and lines $t=\text{const}$ are lines of latitude
$s$ is $\phi$ and $t$ is $\theta$ in spherical polar coordinates

## normals to any surface
$\frac{\partial \vec{r}}{\partial s}$ is tangent to $t=\text{const}$ and $\frac{\partial \vec{r}}{\partial s}$ is tangent to $s=\text{const}$
because $\vec{a}\times \vec{b}$ is perpendicular to both vectors, $\frac{\partial \vec{r}}{\partial s} \times \frac{\partial \vec{r}}{\partial t}$ is the normal to the surface, as is $\frac{\partial \vec{r}}{\partial t} \times \frac{\partial \vec{r}}{\partial s}$

### example: normal of a cylinder
$\vec{r}(s, t) = a\cos(s)\vec{i} + a\sin(s)\vec{j} + t\vec{k}$, $-\pi \leq s < \pi$, $0\leq t \leq h$

$\frac{\partial \vec{r}}{\partial s} = -a\sin s \vec{i} + a \cos s \vec{j}$
$\frac{\partial \vec{r}}{\partial t} = \vec{k}$

$\frac{\partial \vec{r}}{\partial s} \times \frac{\partial \vec{r}}{\partial t} = \left| \begin{align} \vec{i} && \vec{j} && \vec{k}\\ -a\sin s && a\cos s && 0 \\ 0 && 0 && 1 \end{align}\right| = a\cos s\vec{i} + a\sin s\vec{j}$
# areas
The area of an infinitesimal element is $dA = \left|\frac{\partial \vec{r}}{\partial s} \times \frac{\partial \vec{r}}{\partial t} \right|ds dt$
the area of a surface is the infinite sum of the infinite areas
for $a\leq t\leq b$ and $c\leq s \leq d$, 
$\int^d_{s=c} \int^b_{t=a} dA$  

area of a cylinder is $2\pi a h$
area of a sphere of radius $a$ is $4\pi a^2$

# flux integrals
these measure the flow of a vector field $\vec{F}$ through a surface $S$
it is given by $\int\int_s \vec{F}\cdot \hat n dA$

$d\vec{S} = \hat n dA$ 

$\int\int_S \vec{F}\cdot d\vec{S} = \int\int_s \left[\vec{F}(\vec{r}(s, t))\cdot \left( \frac{\partial \vec{r}}{\partial s} \times \frac{\partial \vec{r}}{\partial t} \right) \right]dsdt$